Line Equation

A line is the infinite extension of the shortest connection between two points. Visually, a line is an infinitely long, straight line. There is always exactly one straight line between two points.

All straight lines can be represented by a linear equation, therefore straight lines are also called linear functions .

This article deals with straight lines in ordinary analysis. For straight lines in analytic geometry see: Article on the subject

General line equation

To set up the straight line, you only need the slope and the intersection of the straight line with the $y$ -axis.

\displaystyle y=\textcolor{ff6600}{m}\cdot x+\textcolor{009999}{t}

In this equation, $\textcolor{ff6600}{m}$ is the slope of the straight line and $\textcolor{ff6600}{m}$ is the $y$ -value where the straight line intersects the $y$ -axis.

Constituents of a line equation

A straight line equation consists of a gradient and the $y$ -axis intercept $t$ . These components are explained in more detail below.

As an example consider the line:

\displaystyle y=2x+3

Slope/Gradient

The slope indicates how steep a straight line climbs or falls. From the given equation, one can read out the gradient $m = 2$ .

If you did not know this, you can also determine the gradient using a gradient triangle. To do this, you need at least two different points, which you can obtain by inserting different $x$ -values.

The $y$ -axis intercept $t$

The $y$ -axis intercept $t$ indicates the $y$ -value at which the straight line intersects the $y$ -axis. The value is also obtained by inserting zero for $x$ into the equation of the straight line, since $m\cdot x$ is omitted for the case $x = 0$ and only $y = t$ remains from the original equation.

The fact that the $y$ -axis intercept $t$ has the value 3 in the example can also be seen in the drawing from the line intersecting the $y$ -axis at point $B$ . $B$ has the coordinates $\left(0\left|3\right.\right)$ .

Calculate the equation of a line through two points

Example: Given are the points $A (- 1 ∣ 1)$ and $B (2 ∣ 3)$ . Calculate the equation of the straight line that passes through $A$ and $B$ .

Calculate the slope using the difference quotient $m=\frac{\;\;1-3}{-1-2}=\frac{-2}{-3}=\frac23$
Plug $m$ and any point into the equation of the line to determine $t$ . We use the point $B$ .
$\def\arraystretch{1.25} \begin{array}{l}\begin{array}{ccccc}\mathrm y&=&\mathrm m\cdot\mathrm x+\mathrm t&&\\3&=&\frac23\cdot2+\mathrm t&\;\;\;\;\;\;\;\;\;\;\left|-\frac43\right.&\;\\3-\frac43&=&\mathrm t\;\;\;\;\;\;\;\;&\;\;&\;\end{array}\\\begin{array}{ccc}\;\;\;\;\;\;\;\mathrm t&=&\frac53\;\;\;\;\;\;\;\end{array}\\\end{array}$
Substitute $m$ and $t$ into the general equation of a line. $\;\;\Rightarrow\;\;\mathrm y=\frac23\mathrm x+\frac53$

Calculate the equation of the straight line given the $y$ -axis intercept $t$ and a point $P$ .

Example: Given are the $y$ -axis intercept $t = - 3$ and the point $P (2 / 1)$ . Calculate the corresponding equation of a straight line.

1. Insert $t$ and the coordinates of the point $P$ into the general line equation and solve for $m$ .

\displaystyle \def\arraystretch{1.25} \begin{array}{l}\begin{array}{ccccc}\ y&=&\mathrm m\cdot\mathrm x+\mathrm t&&\\1&=&\mathrm m\cdot2-3&\left|+3\right.&\;\\1+3&=&\;\;\;\;\;\;\mathrm m\cdot 2\;\;\;\;\;\;\;\left|:2\right.&\;\end{array}\\\begin{array}{ccc}\;\;\;\;\;\;\;\mathrm m\;=\;2\;\;\;\;\;\;\;\end{array}\\\end{array}

2. Substitute $m$ and $t$ into the general equation of a straight line $\Rightarrow \;\;y=2x-3$

General lines (interaktive)

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Special lines

Lines through the origin

Such a straight line always has the equation $y=m\cdot x$ , since $t = 0$ holds.

A line through the origin may also represent a direct proportionality.

Constant function

A line that is parallel to the $x$ -axis has the form $y = c$ and is the line of a constant function because it always takes the same, constant value.

Vertical line

A line parallel to the $y$ -axis does not correspond to a function (see definition of a function), but to a relation. It cannot be described with the general equation of a straight line because the gradient would be infinite.

An equation for a vertical line has the form $\mathrm x=\mathrm c$ .

The identity line

The line through the origin corresponding to the function $y = x$ is called the identity line. It is the bisector of the first and third quadrants of the coordinate system.

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